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Bellmore-Merrick Central High School District8th Grade NYS Common Core Learning Standards

Mathematics Curriculum

Written byDiane Coviello, Joan Kleinman, Isabel Raphael, & Jody Tsangaris

June 2012

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Bellmore-Merrick Central High School District8th Grade Mathematics Curriculum

This curriculum was created using an Understanding by Design format. Topics and content are organized under Big Ideas to help students transfer understandings.

Contents of each unit:

Big Ideas, Big Questions, Topic, and Suggested Time Goals Common Core Learning Standards for Mathematics Common Misunderstandings and/or Confusing Concepts Related Seventh-Grade Standards, Skills/Prior Knowledge Vocabulary Additional Resources Understandings and Essential Questions What students will know and be able to do – With Mathematical Practices embedded Corresponding Textbook Pages

Addendum: Analysis of Common Core Learning Standards based on Progressions for the Common Core State Standards in Mathematics by the Common Core Standards Writing Team and posted on http://commoncoretools.me/tools

Textbooks: Mathematics Course 2, Holt, Rinehart and Winston, 2008Mathematics Course 3, Holt, Rinehart and Winston, 2008Amsco’s Integrated Algebra I, Ann Xavier Gantert, Amsco School Publications, 2007

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http://commoncoretools.me/tools

Contents Unit Page1. In Search of Truth: Expressions and Equations 5

Topic: Algebra – Review (integers and two step equations) and Extension with emphasis on solutions of x=a, a = a, and a = b, rational coefficients, application of distributive property and like terms

2. Systems: Number Sets 7 Topic: Number Sets (emphasizing rational and irrational distinction), Square and cube roots and Rational approximations

3. Systems: Scientific Notation 10Topic: Integer exponents, Powers of 10 and Scientific Notation

4. Formulas: Pythagorean Theorem 14Topic: Pythagorean Theorem – Proof of and problem solving. To include distance between any two points on a coordinate plane

5. Formulas: Volume 17 Topics: Volume cylinders, comes and spheres and problem solving

6. Relationships: Parallel Lines 19 Topics: Geometry – Parallel lines, transversal, angle sum, triangles exterior angles, angle-angle similar triangle proof

7. Change: Transformations 21 Topics: Transformations – Development of properties, Sequencing steps, Effect on coordinates, Congruency and Similarity

8. Functions: Visual Functions 25Topic: Functions – defined and graphed, compare representations (algebraic, graphic, tables, & verbal), equation of a line, rate of change and initial value ( slope and y-intercept ), similar triangles to prove slope, distance-time, analyze and sketch graphs given verbal relationships

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9. Functions: Symbolic Functions 31Topics: Functions - Systems – algebraically and with graphic estimation, real world problems in two variables

10. Data: Statistics and Probability 34Topic: Statistics and Probability – Scatterplots and patterns, informal creation of a “line of best fit”, interpret slope and intercept to problem solve, tables to display and interpret patterns - frequency and relative frequency

Addendum – Analysis by Standards, Clusters and Domains 39

Mathematical Practices (full description) 51

Mathematical Practices 1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

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Big Idea: In Search of TruthBig Question: What makes this true?Topic: Expressions and EquationsSuggested Time: 7-10 days

Goals - Common Core Standards:

8EE7: Solve linear equations in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a=b results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Common Misunderstandings/Difficult Concepts:

Inappropriate use of inverse operations Procedural errors Integer Rules Combining like terms Expression vs. Equation Balancing an equation

Related Seventh-grade StandardsSkills/Prior Knowledge:

Solving multi-step equations Knowledge of properties and inverse operations Integer Rules

Additional Resources:

http://www.studygs.net/equations.htmhttp://regentsprep.org/Regents/math/ALGEBRA/AE2/LSolvEq.htmhttp://www.kutasoftware.com/free.htmlhttp://www.math-drills.com/algebra.shtml

Vocabulary:

Commutative property, distributive property, inverse operation, integers, multiplicative inverse, equation, expression, variable, combining, like terms, identity, simplify

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http://www.math-drills.com/algebra.shtml

http://www.kutasoftware.com/free.html

http://regentsprep.org/Regents/math/ALGEBRA/AE2/LSolvEq.htm

http://www.studygs.net/equations.htm

ContentGoals

UnderstandingsStudents will understand that:

Essential Questions

KnowStudents will know:

DoStudents will be able to:

Textbook:

8EE7…a number can be represented with a variable

…algebra can be applied to real word situations

…the order of operations is a universal way of solving a mathematical equation

…when solving equations the property of equality must be adhered to

…no solution, one solution, or multiple solutions may provide “truth” to a given situation

Does “x” always equal the same number?

How can algebra be used to relate to real world scenarios?

Why use algebra to represent a real world scenario?

Should a student in Japan get the same answer as a student in the United States?

What happens when children of different weights get on a seesaw?

How do you isolate the variable?

How is evaluating an expression similar to checking an equation?

a value can be substituted for a variable to solve an equation

when solving real world problems:

to represent an unknown given a verbal problem

there are relationships to symbolically represent

the order of operations

the property of equality

like terms

properties to include the distributive property

a systematic approach to solving a multi-step equation

algebraic equations can be solved by using inverse operations

the difference between an expression and an equation

There is only one solution in this equation: x + 5 = 3x - 3There are an infinite number of solutions in this equation: 1 (x ) = xThere is no solution in this equation:x + 5 = x + 1

check to see if their solution(s) is correct by substituting a number(s) into the variable MP1

represent a given description of a real world situation algebraically MP4

apply the order of operations to check to see if their solution is true MP1

justify each step of simplifying an expression when checking their solution MP1

”keep the balance” when solving an equation MP7

combine like terms solve multistep equations

using properties and inverses or inverse operations MP7

explain each step of solving a multistep equation using correct vocabulary MP1

check the solution of an equation MP1

Given an equation, tell whether or not the equation has one solution, no solution, an infinite number of solutions

Holt Course 3:Chapter 1Pgs. 34– 43

Chapter 11Pgs. 584-598

Pg. 804

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Big Idea: SystemsBig Question: Who am I? Topic: Number Sets (Emphasizing rational and irrational distinction), square and cube roots and rational approximations.Suggested Time: 9 – 12 days


8NS 1 & 2 Know there are numbers that are not rational and approximate them by rational numbers.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2).

For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

8EE 22. Use square root and cube root symbols to represent solutions to equations of the form and , where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.Common Misunderstandings/Difficult Concepts: Full understanding of rational numbers (difference between natural, whole,

integers as well as repeating and terminating decimals). Representing repeating decimals Knowledge of perfect square roots vs. non-perfect square roots Difference between non-terminating decimals vs. repeating decimals Converting terminating decimals to fractions Expressing basic fractions as decimals Understanding of what a square root is (square root of 20 is not 10. Students

frequently divide by two to find square root). Locating basic fractions and decimals on a number line (knowing that 1.5 is in

between 1 and 2)

Related Seventh-grade StandardsSkills/Prior Knowledge: Convert a rational number to a decimal using long

division A rational number terminates in 0s or eventually

repeats Knowledge of rational numbers on an integer number

line The approximate decimal value of pi.

Additional Resources: Vocabulary:

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http://www.mathsisfun.com/irrational-numbers.htmlhttp://www.basic-mathematics.com/converting-repeating-decimals-to-fractions.htmlhttp://www.mathsisfun.com/numbers/cube-root.htmlhttp://www.factmonster.com/ipka/A0876704.html

Cube Root, irrational numbers, perfect cube, perfect square, principal square root, rational numbers, repeating decimals, square roots, terminating decimals, base, exponent, real numbers, whole numbers, integers, natural numbers, non-terminating, approximating, rounding, estimating, power, pi, convert.

ContentGoals


Essential Questions



Textbook:

8NS 1…all rational numbers can be represented as a fraction or a terminating and/or repeating decimal…non-terminating, non-repeating decimals are irrational numbers…rational and irrational numbers can be found on the integer number line

Can numbers always be represented in multiple ways?

How far can numbers go?

definition of irrational numbers a systematic approach to converting a repeating decimal to fractions define the set of rational numbers irrational numbers complete the set of real numbers

show (prove) a number is rational by giving its decimal form which either terminates or eventually repeats MP3

given a repeating decimal expansion, convert it back to a rational number (fraction form) MP8

Holt Course 3:Chapter 2 Pg. 61 – 68

8NS 2…irrational numbers can be approximated without a calculator …the set of real numbers can all be found on the number line

How far can numbers go?

Can an irrational number be placed on a finite number line?

squares to the square root of an imperfect square lies between the perfect square before and after to select decimal values of approximations based on proximity to the perfect square (i.e. is closer to the than

. Initially is

estimate the value of an irrational number MP2

continue to approximate decimal values for irrational numbers

experiment with and explain how to get increasingly closer approximations to the values of irrational numbers MP1,2,6

locate all types of irrational numbers on a number line to include etc. MP4

Holt Course 3:Chapter 2 Pg. 61 – 68

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http://www.factmonster.com/ipka/A0876704.html

http://www.mathsisfun.com/numbers/cube-root.html

http://www.basic-mathematics.com/converting-repeating-decimals-to-fractions.html

http://www.basic-mathematics.com/converting-repeating-decimals-to-fractions.html

http://www.mathsisfun.com/irrational-numbers.html

between 3 and 4. Closer to 3, so between 3.1 and 3.2 and so on) operations with decimals

describe the set of real numbers to include examples and explanation MP3

compare and contrast rational with irrational numbers

ContentGoals


Essential Questions



Textbook:

8EE 2…inverse operations can be used to evaluate perfect square roots and perfect cube roots.

What makes a square “perfect?”

Is there a difference between finding the square of a number and its square root?

Is there a difference between finding the square of a number and the cube of a number?

What is the difference between square root and cube root?

given , is defined as the positive solution (when it exists) given , the solution may be stated as or the definition of perfect squares and perfect cubesCubes to ? the definition of the set of irrational numbers is irrational, where a is a non-perfect square

evaluate square roots and cube roots of perfect roots (Squares to and cubes to

? ) MP6 & 7 explain why is irrational represent the solution to an

equation in the form of a square root or a cube root MP4

explain why or MP2 & 3

Holt Course 3:Chapter 4Pg. 182-194

Consider the end of each of these chapters for additional review.

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Big Idea: SystemsBig Question: Where do I belong?Topic: Integer Exponents, Powers of 10 and Scientific NotationSuggested Time: 7 – 10 days


8EE 1, 3, & 4 Work with radicals and integer exponents.

1. Know and apply the properties of integer exponents to generate equivalent numerical expressions.

For example, .

3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

For example, estimate the population of the United States as 3 times and the population of the world as 7 times , and determine that the world population is more than 20 times larger.

4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.


A negative exponent yields a negative numberThe exponent in scientific notation is the amount of zeros in the numberA whole number does have a decimalApplying the Laws of Exponents properlyPlace values in decimals


Applying Laws of exponentsInteger Rules


http://www.purplemath.com/modules/exponent3.htmhttp://www.mathsisfun.com/algebra/exponent-laws.html

Vocabulary:

Integers, exponents, equivalent, expressions, fractions, power, estimate, operations, rational, scientific notation.

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http://www.mathsisfun.com/algebra/exponent-laws.html

http://www.purplemath.com/modules/exponent3.htm

ContentGoals


Essential Questions



Textbook:

8EE1

…rules are used for consistency in math.…exponent rules can be discovered through understanding of mathematical abbreviations.

…Negative and zero exponents relate to what happens when the laws of exponents are applied

Can numbers be represented multiple ways?

the basis for the laws of exponents and use them to generate equivalent numerical expressions

how to rewrite numbers with negative exponents

when a whole number has a negative exponent, the value of the power is less than 1

Discover and create the laws of exponents for multiplication and division. MP 7

Multiply and divide exponents with the same base and be able to express them in exponential form, expanded and standard form. MP 7

Evaluate negative exponents by- rewriting them using a

fraction with a positive exponent

- as a decimal. MP 7

Holt Text:Course 3

Chapter 4Pgs. 159-173, 180

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ContentGoals


Essential Questions



Textbook:

8EE 3…very large or very small numbers are commonly written in scientific notation for each of use.

Can the same number have a different look?

the format of a number written in scientific notation

a strategy to translate numbers from scientific notation and standard form and vise versa

know the meaning of the exponents to compare numbers written in scientific notation

Translate numbers from scientific notation and standard form and vise versa. MP 7

Compare and order numbers written in scientific notation. MP 6

Analyze and work with numbers in scientific notation within verbal problems and/or charts. MP6

Use a computer or calculator to generate numbers in scientific notation, interpret their meaning

Holt Text:Course 3

Chapter 4Pgs. 174 – 178, 180

8EE4…numbers can be represented in scientific notation.

…operations can be performed on numbers written in scientific notation.

How do you add, subtract, multiply, and divide using scientific notation?

a method for performing all four operations using scientific notation

what to consider when choosing units of appropriate size for measurements of very large or very small quantities

that any expression written in scientific notation can be manipulated to perform operations using properties

Add numbers written in scientific notation. MP 6

Subtract numbers written in scientific notation. MP 6

Multiply numbers written in scientific notation. MP 6

Divide numbers written in scientific notation. MP 6

Create equivalent expressions of numbers written in scientific notation (see example under 8EE4)

Holt Text:Course 3

Chapter 4Pgs. 179 – 180

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Big Idea: FormulasBig Question: What am I looking for?Topic: Pythagorean Theorem – Proof of and problem solving. To include distance between any two points on a coordinate plane.Suggested Time: 7 - 10 days


8G 6-8 Understand and apply the Pythagorean Theorem.

6. Explain a proof of the Pythagorean Theorem and its converse.

7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.


Try to use Pythagorean Theorem with angles Identifying the parts of a right triangle accurately Applying Square Roots Squaring a number Solving algebraically to find the length of a missing side

Related Seventh-grade StandardsSkills/Prior Knowledge: Solving Equations Evaluating Exponents Square Roots Identifying Right Triangles Coordinates of Points Plotting points on the coordinate plane

Additional Resources:http://www.mathsisfun.com/pythagoras.htmlhttp://www.cut-the-knot.org/pythagoras/index.shtmlhttp://www.mathopenref.com/coorddist.html

Vocabulary:Pythagorean Theorem, Exponents, Square Roots, Equations, Converse, Hypotenuse, Right Triangle, Distance Formula, Legs

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http://www.mathopenref.com/coorddist.html

http://www.cut-the-knot.org/pythagoras/index.shtml

http://www.mathsisfun.com/pythagoras.html

ContentGoals


Essential Questions



Textbook:

8G6

…the Pythagorean Theorem mathematically describes the relationship of the sides of a right triangle

What is the relationship of the sides of a right triangle?

Can you prove the Pythagorean Theorem and its converse?

How can looks be deceiving? Should our judgments be based on what we see?

the characteristics of the parts of a right triangle (legs, hypotenuse, right angle)

the relationship between the lengths of the sides of a right triangle as a2+b2=c2, where a and b are the legs and c is the hypotenuse

given a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse

if the sum of the squares of the legs of a triangle equal the square of the hypotenuse, the triangle is a right triangle.

a proof explaining Pythagorean Theorem

how to calculate the area of squares

create and explain the proof for the Pythagorean Theorem – use area formulas MP3

sketch and label a right triangle with a, b, and c and/or any given measurements in various orientations MP2

Record, substitute and solve for unknowns using the Pythagorean Theorem

calculate square root using a calculator or approximate it and round appropriately. MP5

apply the appropriate units of measurement

explain under what conditions a triangle can be proven to be a right triangle

use the Pythagorean Theorem to prove that a triangle is a right triangle MP3

Holt Course 3:Chapter 4Pgs. 195 - 200

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ContentGoals


Essential Questions



Textbook:

8G7…the Pythagorean Theorem is a tool that can be used to find the length of an unknown side in a right triangle, given two sides

How can you use the Pythagorean theorem to solve problems?

to use the Pythagorean Theorem when information given includes two sides and the length of a third is the question.

to sketch a drawing from the verbal problem and label it with the given information

to recognize the right angle

interpret a verbal problem with information indicating the use of the Pythagorean Theorem to find a missing length MP1

sketch a drawing representing the given information and identify the sides of the triangle MP7

use the Pythagorean Theorem, substitute and solve for the missing side with appropriate units.

Holt Course 3:Chapter 4Pgs. 195 - 200

8G8

…the Pythagorean Theorem can be used to find the length of a segment on the Coordinate Plane(leads into an understanding of rise and run and the distance formula)

How can you find the actual length of a line on coordinate plane when it is not parallel to the x or y axis?

that for any diagonal line on a coordinate plane, lines can be drawn from endpoints parallel to the x and y axis creating a right angle at the intersection

the length of the new lines can be determined by the units on the coordinate plane or by calculating the length subtracting coordinates

Given a line on the coordinate plane, sketch a right triangle and use the coordinates to label it.

use the Pythagorean Theorem to find the length of the missing side MP1

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Big Idea: FormulasBig Question: What do I need? What am I looking for?

Topic: Volume Cylinders, cones and spheres and problem solvingSuggested Time: 5 – 7 days


8G9 Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.


Pi = 3.14 Radius vs. Diameter Area vs. Circumference Volume vs. Surface Area Labeling appropriate units Rounding Difference between Estimating and Rounding


Area of two dimensional figures Parts of a Circle Area and Circumference of a Circle Volume of right prisms


http://www.mathsteacher.com.au/year9/ch14_measurement/18_cylinder/cylinder.htmhttp://math.about.com/od/formulas/ss/surfaceareavol_2.htmhttp://www.aaastudy.com/exp79_x8.htm

Vocabulary:

Volume, area, circumference, cones, cylinders, spheres, three-dimensional shapes, radius, height, base, pi, rounding, estimating, powers, units.

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http://www.aaastudy.com/exp79_x8.htm

http://math.about.com/od/formulas/ss/surfaceareavol_2.htm

http://www.mathsteacher.com.au/year9/ch14_measurement/18_cylinder/cylinder.htm

ContentGoals


Essential Questions



Textbook:

8G9

…formulas can be used to find the volume of a sphere, cone, and cylinder

How can you solve problems using the formulas for volume?

some ideas about how the formulas for volume of cones, cylinders and spheres was derived

how to recognize which formula is appropriate for a given situation and given information

the difference between perimeter, area, surface area and volume

how to recognize the correct values for the volume formulas

Analyze given information to determine what is being asked

Sketch or label a drawing with measurements

Calculate the volume of cylinders, cones and spheres. MP7

Apply the appropriate unit of measurement for surface area. MP6

Justify the use of the appropriate formula. MP6

Evaluate any formula given certain values. MP7

Holt Course 3:Chapter 8Pgs. 412 – 426, 436 – 439, 444, 450 - 453

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Big Idea: Relationships Big Question: What’s the relationship?Topic: Geometry- Parallel lines, transversal, angle sum, triangles- exterior angles, angle-angle similar triangle proof Suggested Time: 7 – 10 days


8G55. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.


Correctly identifying angle relationships and angle measures Using correct terminology to define angle relationships


Supplementary Angles Angle facts Types of triangles Vertical Angles Sum of Interior angles in a triangle


http://www.mrperezonlinemathtutor.com/G/1_2_Angles_in_Parallel_lines.htmlhttp://www.regentsprep.org/regents/math/geometry/GP8/Lparallel.htmhttp://www.core-learning.com/downloads/resources/math/geo_apdx2.pdf

Vocabulary:

Angles, parallel lines, transversal, alternate interior, alternate exterior, vertical angles, corresponding, same side interior angles, supplementary angles, interior angles, exterior angles, remote interior angles, triangle, similar, congruent, relationships, proportion

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http://www.core-learning.com/downloads/resources/math/geo_apdx2.pdf

http://www.regentsprep.org/regents/math/geometry/GP8/Lparallel.htm

http://www.mrperezonlinemathtutor.com/G/1_2_Angles_in_Parallel_lines.html

ContentGoals Understandings

Students will understand that:

Essential Questions



Textbook:

8G5…different angles are formed by a transversal that intersects two parallel lines.

…patterns are formed when a transversal intersects two parallel lines.

…relationships are created by angles and triangles formed by intersecting lines.

…missing angles can be calculated by understanding the relationships between angles created by intersecting lines.

What can you conclude about the angles formed by parallel lines that are cut by a transversal?

What can you conclude about the measures of the angles of triangle?

How can you determine when two triangles are similar?

the names and relationships of angles represented when parallel lines are intersected (cut) by a transversal

the sum of the interior angles of a triangle is 180°

the sum of the two remote interior angles of a triangle is equal to the exterior angle of the third angle in that same triangle.

the sum of the angles in a triangle is equal to the sum of any interior angle and the exterior angle adjacent to it.

two triangles that have two congruent angle measurements are similar triangles. (angle-angle theorem)

given similar triangles, the lengths of corresponding sides are proportional

how to determine if two triangles are congruent based on given information about angles and/or length measurements

Recognize a drawing with accompanied explanation as parallel lines cut by a transversal. MP3

Label drawings by applying relationships of angles

Determine through observation, interaction with, or any other method the relationships between angles given parallel lines cut by a transversal. MP3

Determine the measurement of a missing angle in a triangle given facts (such as two interior angles, classification of the triangle, measure of the exterior angles, etc). MP2

Calculate the length of missing sides of a triangle given similar triangles. MP1

Prove two triangles are similar given length measurements and/or angles. MP1

Holt Text Course 3 Chapter 7 pgs. 330 – 333, 336 – 340, 352 – 357

Chapter 5 pgs. 236 - 241

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Big Idea: ChangeBig Question: Does change produce something new?Topic: TransformationsSuggested Time: 2 weeks

Goals - Common Core Standards: Geometry:

8G. 1- 4 Understand congruence and similarity using physical models, transparencies, or geometry software.

1. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length.b. Angles are taken to angles of the same measure.c. Parallel lines are taken to parallel lines.

2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.


Confuse the term translation and transformation Rotate counter-clockwise versus clockwise Reflecting over the wrong axis

Related Seventh-grade StandardsSkills/Prior Knowledge:7.G Draw construct, and describe geometrical figures and describe the relationships between them.Work with scale drawings and similar figures.Properties of geometric figures.7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems.Work with proportions.

Additional Resources:Jmap.org appleseed analyticsregentsprep.org Khan institutebrainpopHolt onlinebrightbulb.com

Vocabulary:Transformation, pre-image, image, congruent, similar, isometry, translation, rotation, reflection, dilation, composition of transformations

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ContentGoals


Essential Questions



Textbook:

8G1 …a transformation is an operation that maps or moves the points of a figure in a plane

…each transformation has a unique set of properties

What changed?

What’s preserved and when?

the term transformation describes all the different changes

the definition and properties of a rotation, reflection, translation

properties of polygons (angle and side relationships)

properties of polygons preserved under a rotation, reflection or translationa. distance is preserved (length of segments are the same)b. angle measures are preserved ( remain the same)c. parallel lines remain parallel

proper labeling notation (primes)

verify experimentally the properties of rotations, reflections, and translations MP5

observe and generalize about what changed when a figure is transformed MP1

accurately identify and explain properties of each transformation MP1

accurately identify and explain properties of a polygon that are preserved under each transformation MP2

accurately identify and explain the properties of a polygon being preserved and not preserved MP2

graph rotations, reflections, and translations with proper labeling on the coordinate plane MP5

HOLT Mathematics Course 3 p. 358-363

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ContentGoals


Essential Questions



Textbook:

8G2... a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations

How can a congruent figure be created?

the definition of congruent figures

that a composition of transformations can form an image that is congruent to the pre-image

describe a sequence (of rotations, translations and/or reflections) that exhibits congruence between two given congruent figures, MP7

sketch the sequence of transformations to create a congruent two dimensional figure

8G3…formulas may be used to accurately identify points of transformation

...a dilation has a unique set of properties …whether the properties of a given polygon are preserved is dependent upon the transformation

How can you accurately change the position or size of a figure?

What’s preserved and when?

the definition and properties of a dilation

procedures for drawing each transformation applying a general rule T(a,b) (x,y) → (x + a, y + b)R90 (x,y) → (-y,x)R180(x, y) → (-x, -y)r y-axis (x, y) → (-x, y)r x-axis (x, y) → (x, -y)r origin (x, y) → (-x, -y)Dk(x, y) → (kx, ky)

graph a dilation with proper labeling on the coordinate plane MP5

describe the effect of dilations, translations, rotations, and reflections using coordinates MP1

create a general rule for each transformation using this effect MP5

describe how to graph each transformation MP3

compare and contrast the different procedures for each transformation MP4

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ContentGoals


Essential Questions



Textbook:

8G4.. a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations

How can a similar figure be created?

the properties of similar figures

that a composition of transformations can form an image that is similar to the pre-image

describe a sequence that exhibits the similarity between a given pair of two-dimensional similar figures MP4

sketch the sequence of transformations to create a similar two dimensional figure

Big Idea: FunctionsBig Question: What fits? What works?Topic: Visual FunctionsSuggested Time: 4- 6 weeks


23

8F 1-5 Define, evaluate, and compare functions.

1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1

2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Use functions to model relationships between quantities.

4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally._________________1Function notation is not required in Grade 8.

8EE 5 & 6

Understand the connections between proportional relationships, lines, and linear equations.

5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

24

6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.


Confusing input with output Creating an equation from a table of values Writing an equation from a word problem


7A7, 7A10


Jmap.orgRegentsprep.orgHoltonlineKhan Institute

Vocabulary:

Domain, range, function, relation, input, output, independent variable, Dependent variable, slope, y intercept, direct proportion, ordered pair

Use unit rate, slope, rate of change and constant of proportionality interchangeably

ContentGoals


Essential Questions



Textbook:

8F1…a function is a special relation

Why is a given relation a function?

definitions of the terms: domain, range, relation, function, rate of change, initial valuean equation with two variables defines a relationship

create a mapping diagram for a relation (MP4) pgs. 134-

137

8F3

…relations can be represented in various formats

When presented with a function in a table or graph, how do you list the ordered pairs represented?

how to read a table of values

how to apply the vertical line test

identify the domain and range

identify a function, given a relation

identify a function given a graph

identify a function given a

pgs. 134-141

25

…not all functions are linear

How do you write an equation?

the characteristics of a function that is not a line

verbal description explore and describe non-

linear function

8EE5

With 8F2

…two different functions can be compared using two different methods

…a visual representation of a proportional relationship is its graph

Can you compare a function represented by a graph to a function represented by an equation or a table of values?

the constant of proportionality: a unit rate is slope on a graph

a real world situation can be expanded and graphed. D = 60t

it is possible to compare the rate of change for 2 different functions represented in different ways

using proportions, create equivalent rates and graph

compare all the forms of the proportional relationship: equation, unit rate (slope), table, graph.

make observations about the proportional relationships graphed

identify and verbally describe the rate of change for 2 different functions that are represented in different ways for comparison purposes. ie. One in the form of a table and the other as an equation.(MP1)

pgs. 133-140

ContentGoals


Essential Questions



Textbook:

8F2 with 8EE5

…an equation can be written to represent the situation in a word problem

How can you graph the situation portrayed in a word problem?

Can you identify a direct proportion from a graph?

how to write an equation in the form y = mx

it is possible to compare 2 different proportional relationships represented in different way

definition of a ratio

definition of a direct proportion using the form:

write an equation from a word problem - graph the equation - find the slope

find the value of “m” for a direct proportion written as an equation or in a table

find the value of “m” on a graph - compare the two situations

26

y = mx

8EE6

…the steepness of a line is its slope

…slope is rise/run

…the “y” intercept is the point at which the graph intersects the y-axis

Can you identify a line with positive, negative, zero and undefined slopes?

Does the slope of a line change depending on the 2 points you choose to determine the slope?

how to find the slope of a line given the graph of the line

how to construct similar triangle from 3 points on a line

the graph changes at a constant rate

how to write the equation of a line by knowing the slope and y-intercept

find the slope of a line, given its graph

write the ratios of the vertical side length to the horizontal side length for each similar triangle created when graphing 3 points on the line

compare each ratio to the slope of the line

identify the y-intercept of a line from a table of values. (MP1)

find the slope of a line given 2 points on the line

write the equation of a line given the slope and a point passing through the line

pgs. 633-641

638-643

644-646

27

ContentGoals


Essential Questions



Textbook:

8F4…functions can be used to model the relationship between 2 quantities

…linear functions are used to solve real world problems

Why is the “y” value the dependent variable?

Why is the “x” value the independent variable?

How can you interpret the graph of a linear function to represent real life situations?

given a table, how to find the missing output value(s)

given a table of values or a graph, how to determine the equation for a function

the rate of change is the slope

a graph can represent the rate at which 2 quantities are changing

write an equation from a table of values or a graph

fill in the output values from a table of values

determine the rate of change and the initial value

write an equation, given a real life situation. Construct a table of values for your equation

identify and interpret the initial value and rate of change MP4

pgs. 125

633-636

637-640

8F5

…a graph can be broken down into its parts as it changes

How can you represent a sequence of events on a graph?

a graph increasing at a constant rate will be represented by a line segment with an upward slant

a graph decreasing at a constant rate will be represented by a line segment with a downward slant

a horizontal line segment on a graph represents no change

given a real world situation that represents a sequence of events, graph the function piece by piece MP4, MP2

analyze a graph and describe in words the relationship between the two quantities

pgs. 128-131

28

Big Idea: FunctionsBig Question: How can I represent my unknown variables?Topic: Symbolic Functions ( Systems of Equations-Algebraically with graphic estimation, real world problems in 2 variables )

Suggested Time: 1-2 weeks


8EE8

8. Analyze and solve pairs of simultaneous linear equations.

a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Relates to 8EE7: x=a a=a, a=b

For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two variables.

For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.


Using 2 variables to solve a problem



Regentsprep.orgJmap.orgKhan instituteHoltonline

Vocabulary:

Substitution, elimination of a variable, coefficient, consistent equations, coinciding, simultaneous equations,

29

ContentGoals


Essential Questions



Textbook:

8EE8…definitions of intersecting, parallel and coinciding lines

…a system of linear equations can be 2 or more equations with the same variables

…a system of linear equations can be solved graphically

…reasoning skills can sometimes be utilized to solve a system of linear equations

…linear equations can be used to solve real life problems when given 2 different relationships with 2 unknown variables

What are the differences between intersecting, parallel and coinciding lines?How many solutions can a system of equations have?

What are the methods of solving a system of equations algebraically?

How can reasoning skills be applied to find the solution to a system of linear equations?

Can you write two equations to represent the situation in a problem using two different variables?

intersecting lines have different slopes

parallel lines have the same slopes

coinciding lines have the same slopes and y intercepts

intersecting lines will have one point in their solution

parallel lines will not have any points in their solution

coinciding lines will have the same points in their solution

substitution is a method to solve a system of linear equations

elimination of a variable is a method to solve a system of linear equations

given pairs of linear equations, determine if the pairs represent intersecting, parallel or coinciding lines by comparing the slopes and y intercepts MP1

solve given systems of equations graphically and check their solutions algebraically MP4

compare given equations to see if you can use reasoning to find the answer - for example, 4x + 2y = 64x + 2y = 12-there is no solution, because 4x + 2y cannot simultaneously equal 6 and 12

represent the variables in a word problem - write and solve a system of linear equations - Check your answers

608-610

666-667

611

30

Continued on next page some solutions will not have integer values- estimation can be used to approximate the answer by finding a point that is nearby on the graph of the line

you must represent and define the variables you choose (explain what the variables stand for)

31

Big Idea: DataBig Question: What can you interpret?Topic: Statistics and ProbabilitySuggested Time: 9 days

Goals - Common Core Standards: 8SP1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

Common Misunderstandings /Difficult Concepts:

Students draw a line of best fit near outliers instead of where data clusters Students confuse dependent and independent variables


Students have made predictions with data on populations

Predicting probabilities based on frequency of data


Regentsprep.org khan academymathbits.comalbany.eduilluminations.nctm.org

Vocabulary:

Bivariate data, independent variable, dependent variable, outlier, correlation, linear and non-linear associations, line of best fit, slope, y-intercept, scatter plot, frequency table, categorical data, two-way table

32

ContentGoals


Essential Questions



Textbook:

8SP.1 …scatter plots are a pictorial display of bivariate data

…conclusions can be drawn about the relationship (or lack thereof) between the variables

… correlation does not always indicate causation

How can a scatter plot be used to draw inferences about the relationship between two variables?

What is the form of the relationship?

what a scatter plot is and when it is used to represent and analyze bivariate data

a method for constructing and interpreting scatter plots

correlations (positive, negative, or none) and linear/non-linear associations

vocabulary to describe patterns

given 2 sets of data (bivariate), construct a scatter plot MP4

classify one set of data as the dependent variable (y) and one set as the independent variable (x) MP1

describe any patterns, such as identifying outliers, clusters and any correlation (positive, negative, or none) MP7

draw conclusions about the correlation of the set of data MP3

investigate patterns of a linear or non-linear association MP7

HOLT Mathematics Course 3p. 494-517

AMSCO'S Integrated Algebra Ip. 710-728

33

ContentGoals


Essential Questions



Textbook:

8SP.2…straight lines can model relationships between two quantitative variables

…the absence of a linear association does not mean that the variables are unrelated

What is the form of the relationship?

How can predictions be helpful?

the line should be the best representation of the relationship, passing through some, none, or all of the points

the line of best fit is also called a trend line

the line of best fit is useful for predicting values that are not in the plot

determine if a line can be used to describe the data MP3

determine the best location to graph a line that would represent the relationship and fit a straight line MP1

determine the characteristics of a line that indicate a strong relationship and one that would indicate a weak relationship MP4

make predictions about the relationship based on line drawn MP4

8SP.3…the equation for the line of best fit aids in making predictions

…the further you get from the data in the plot, the more unreliable your prediction is

What does the slope reveal about the relationship?

What does the y-intercept tell us about the data?

How can the equation of the line of best fit be used?

a formula can be used to make predictions

slope shows a rate of change (constant of proportionality/unit rates)

the y-intercept represents some initial value

calculate and interpret the slope and y-intercept of the line of best fit MP4

write the equation of the line of best fit MP4

use the equation of the line to interpret data, solve verbal problems, and make predictions MP1

make sense of problems and recognize regularity in repeated reasoning (slope) MP8

34

ContentGoals


Essential Questions



Textbook:

8SP.4

…patterns of association can also be seen in bivariate categorical data

…the interpretation of the data in a two-way table lies in comparing/contrasting the rows and columns

…the calculation and comparison of the percents (relative frequencies) in each category can provide evidence of a possible association

How can percents help to find an association between variables?

categorical variables take a value that is one of several possible categories and have no numerical meaning- Ex.: Hair color, gender, political affiliation

a method to interpret categorical data in a two-way table

construct a two-way table of categorical data by displaying frequencies MP4

use the totals across the rows and down the columns of the two-way table to calculate the percent of each category by row or by column to construct a two-way table of the relative frequencies MP6

analyze and compare the relative frequencies to describe a possible association between the two variables MP2

describe the evidence found to defend a possible association between variables MP2

35

Example of two-way tables:

The two-way table shows the favorite leisure activities for 50 adults - 20 men and 30 women. Because entries in the table are frequency counts, the table is a frequency table.

Dance Sports TV Total

Men 2 10 8 20

Women 16 6 8 30

Total 18 16 16 50

We can also display relative frequencies in two-way tables.


Men 0.10 0.50 0.40 1.00

Women 0.53 0.20 0.27 1.00

Total 0.36 0.32 0.32 1.00

Relative Frequency of Row


Men 0.11 0.62 0.50 0.40

Women 0.89 0.38 0.50 0.60

Total 1.00 1.00 1.00 1.00

Relative Frequency of Column

Each type of relative frequency table makes a different contribution to understanding the relationship between gender and preferences for leisure activities. For example, "Relative Frequency for Rows" table most clearly shows the probability that each gender will prefer a particular leisure activity. For instance, it is easy to see that the probability that a man will prefer dance is 10%; the probability that a woman will prefer dance is 53%; the probability that a man will prefer sports is 50%; and so on.

36

http://stattrek.com/Help/Glossary.aspx?Target=Frequency%20table

Addendum: Analysis of Standards, Clusters, and Domains based on Progressions for the Common Core State Standards in Mathematics by the Common Core Standards Writing Team and posted at commoncoretools.wordpress.com.

8.NS The Number System

Know that there are numbers that are not rational, and approximate them by rational numbers.

1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2).

For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

KNOWStudents will know . . .

DOStudents will be able to . . .

Definition of irrational numbers A systematic approach to converting a repeating decimal to

fractions The square root of an imperfect square lies between the perfect

square before and after To select decimal values of approximations based on proximity to

the perfect square i.e. is closer to the than . Initially is between 3 and 4. Closer to 3, so between 3.1 and 3.2 and

so on Operations with decimals Define the set of rational numbers Irrational numbers complete the set of real numbers

…show a number is rational by giving its decimal form which either terminates or eventually repeats (7th grade)…given a repeating decimal expansion, convert it back to a rational number (fraction form)…estimate the value of an irrational number…continue to approximate decimal values for irrational numbers…experiment with and explain how to get increasingly closer approximations to the values of irrational numbers…locate all types of irrational numbers on a number line to include

etc.

37

8EE: Work with radicals and integer exponents.

1. Know and apply the properties of integer exponents to generate equivalent numerical expressions.

For example, .2. Use square root and cube root symbols to represent solutions to equations of the form and , where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times and the population of the world as 7 times , and determine that the world population is more than 20 times larger.

4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.For example: given that we breathe about 6 liters of air per minute, they estimate that there are minutes per day, and that we therefore breath about liters in a day. Expressions and Equations: Common Core Standards Writing Team 4/22/11



Properties of integer exponents1. 2. 3. 4.

5.

6. Where the properties of integers originate, equivalent forms

leading to the properties of integers

…observe and create the properties of integer exponents…define each property and give examples…fluidly use the properties of integer exponents as well as all the properties to create equivalent forms of expressions (carried over from 7th

grade to add to the complexity of expressions)…validate steps in expanding or simplifying expressions…prove the equivalency of expressions…evaluate square roots and cube roots of all “small” perfect roots…explain why is irrational…represent the solution to an equation in the form of a square root or a cube root…expand the powers of 10…explain the relationship between exponential and expanded forms of

38

Square roots and cube roots are inverses of powers Given , is defined as the positive solution (when it

exists) Given , the solution may be stated as or The definition of perfect squares and perfect cubes The definition of the set of irrational numbers is irrational, where a is a non-perfect square The definition of the exponent in and relate to the expanded

form The relationship of to its decimal equivalent Scientific notation To manipulate any expression written in scientific notation to

perform operations with other numbers written in scientific notation using properties

Consistent labeling throughout operations with appropriate units Conversions within systems may be necessary for a final answer

base 10…given any large number, “estimate” the value of that number in the form of a single digit times a whole-number power of 10. i.e. 5,234,567 would be …given any small number, estimate the value of that number in the form of a single digit times a whole-number power of 10…compare estimated values of very large and/or very small numbers …describe how many times larger one estimated form is to another…take any number and write it as a single digit times a power of 10 (scientific notation)…perform calculations with numbers written in scientific form.…create equivalent expressions re-writing numbers in scientific form to solve problems (see example under standard 4)…apply appropriate units when solving a verbal problem containing scientific notation (see example under standard 4)…generate numbers in scientific notation using computer/calculator technology and interpret the meaning of those numbers

39

Understand the connections between proportional relationships, lines, and linear equations.5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.KNOWStudents will know . . .


Given a term to represent a real world situation, the variable represents a range of values i.e. 60t can represent the distance traveled by a car in t minutes going 60 miles per hour

Proportional relationships can be represented graphically, in a table and in the form of an equation

The coordinate plane, order pairs, plot points The constant of proportionality: A unit rate (the ratio between two

quantities being compared – 7th grade) is the slope of the line on a graph

An equation in two variables defines a relationship Definition of similar triangles Definition of right triangle and parts of the right triangle Ratio of rise to run The equation of a line through the origin is y = mx The equation of a line through the vertical axis at b is y = mx + b

…using proportions create equivalent rates and graph them. (Display in a table and label graph with quantities being compared) -make observations about the proportional relationships graphed -define a unit rate based on the graph and proportional relationships…represent output as a variable. For example, d represents distance when t represents time at 65 miles per hour, …define the relationship between the two variables…develop equations in two variables, defining each variable (informal introduction of a function)…compare all the forms of the proportional relationship, slope: table, graph, and equation…compare two different rates in two different forms (see example under standard 5)…prove that corresponding sides of similar triangles drawn on line in a graph are proportional…explain why the slope m on a line is the same between any two distinct points…write the equation of a line through the origin given the concept that the relationship between the variables is proportional using values from the right triangle drawn with the hypotenuse as the line segment from (0,0) to (x,y) similar to right triangle (0,0) to (m,1)

or leading to …use examples to prove your equation is correct. Substitute a value for x and see if you get y on the graph…explore what happens when the line does not go through the origin…define the effect of a line through (x,b) on the equation of the line

40

Analyze and solve linear equations and pairs of simultaneous linear equations.

7. Solve linear equations in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

8. Analyze and solve pairs of simultaneous linear equations.

a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two variables.

-For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.-For example, Henry and Jose are gaining weight for football. Henry weighs 205 pounds and is gaining 2 pounds a week. Jose weighs 195 pounds and is gaining 3 pounds a week. When will they weigh the same? from Expressions and Equations: The Common Core Standards Writing Team 4/22/11



A systematic approach applying properties for solving complex equations with variables on both sides, like terms, parentheses, and the entire number system for coefficients and constants

The possibility that there might be one solution, no solutions or

…solve complex equations with variables on both sides, like terms, parentheses, and the entire number system for coefficients and constants…explain properties used for the systematic solution of complex equations

41

an infinite number of solutions to an equation The characteristics of an equation with only one solution, no

solution, or an infinite number of solutions When x = one number it has been solved for one solution When a=a there is an infinite number of solutions. The equation

is true for any number substituted for the variable When a=b there is no solution. There is no value for the variable

that will make the statement true The solution to a system of equations is that set of ordered pairs

that make both equations true The solution to a system is the point (s) of intersection on a graph

of the equations To create a table by substituting values and graphing the ordered

pairs To analyze a linear equation for slope and y-intercept and graph

the line starting with the y-intercept and rate of change (slope) A systematic approach to transforming equations into y =’s. If two equations are in the slope y intercept form, they can be set

equal to each other Substitute the value for x into an equation to solve for y. The graph of a system with one solution is two intersecting lines The graph of a system with no solutions is parallel lines The graph of a system with infinite solutions is that the equations

represent the same line – one line which includes all the points on that line

…apply the concept of transforming equations into simpler forms to evaluate them for possible types of solutions (values for the variable that make the statement true)…examine the process of solving various equations leading to one solution, no solutions or an infinite number of solutions that make the equation true…analyze and describe characteristics of equations with one solution, no solution or an infinite number of solutions.…create generalizations about the possible solutions for equations…prove that an equation has one, no or infinite solutions by transforming them into an x=a, a=a, or a=b format…graph linear equations using a table and/or slope-intercept method…graph a system of equations (2 or more) and make observations…substitute the ordered pair at the intersection of the lines into both equations and make observations. Define the solution to a system of equations…transform equations into slope y-intercept form, y = mx + b using inverses and properties…solve a system of linear equations algebraically -when two equations are in slope y-intercept form, y = mx + b, set them equal to each other (if a = b and b = c then a = c) and solve for x. Substitute for one of the two variables to solve for the second. Check with the other equation -by elimination/addition - multiply to get additive inverses for one variable and add to eliminate one variable. …compare algebraic solutions of systems with the solution determined by graphing the lines…examine graphs of equations with one solution, no solutions or an infinite number of solutions and draw conclusions…examine systems of equations set up to be solved algebraically and predict whether there will be one solution, no solutions, or an infinite number of solutions…analyze mathematical scenarios asking for graphical and algebraic evidence regarding sets of points and/or lines on a graph (see the example under standard 8c)…solve real world problems that can be modeled by creating two linear equations in two variables. (see the example under standard 8c)

42

-analyze a given verbal problem for two separate sets of information -determine the relationship between the two sets of separate sets of information - create equations in two variables that can be solved as a system to determine the solution - answer the question to the real world problem using the solution to the system of equations

43

8F Define, evaluate, and compare functions.

1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1

2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

For example, the function giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

1Function notation is not required in Grade 8.KNOWStudents will know . . .


Definition of a function, domain and range, independent and dependent, input is x, output is y

Vertical line test Coordinate plane, ordered pairs, points To substitute values for x to find y creating ordered pairs or a

table of input-output values Slope, m, as the constant of proportionality – unit rate (See 8EE) as a linear function through the origin as a linear function through the y axis at b The characteristics of a line related to the numerical slope – rate

of change (greater or less degree of slope – rate of change, positive or negative, increase or decrease)

Characteristics of equations that are not linear functions Points on a line are solutions of the equation .

…determine whether a relation is a function given an equation, table, set of ordered pairs or graph…given a function create a table of values…graph a function using the table of values…graph a function using the slope intercept method…use appropriate notation when graphing and labeling a line given a function…give a verbal description of a linear function…prove a given point is on a line…examine algebraic representations of linear functions and make comparative observations regarding the structure (slope and y intercept)…examine graphic representations of linear functions and make comparative observations regarding the structure – determine the slope and y intercept…examine linear functions represented on a table and make comparative observations about the structure – determine the slope and y intercept

44

…compare two different linear functions represented in different ways (table, graph, algebraically, description)…compare and discuss rates of change given a variety of examples in different formats…graph relations and functions that are not linear from a table of values…make observations regarding the characteristics of the graphs of functions and relations that are not linear…make observations regarding the characteristics of other types of functions and relations in equation form. …given an equation, state with evidence whether it represents a linear function (transform it to a y= if necessary)

45

8.G Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software.

1. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length.b. Angles are taken to angles of the same measure.c. Parallel lines are taken to parallel lines.2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.



Congruent Similar Properties of rotations Properties of reflections Properties of translations Properties of dilations 2 two-dimensional figures are congruent if one can be rotated,

reflected and/or translated to obtain the second one.

…rotate, reflect and translate figures. Make observations and generalizations about line segments and angles…compare and contrast rotation, reflection and translations…identify corresponding, congruent line segments and angles under a rotation, reflection, or translation (orientation of the figure does not change the measurement of segments or angles)…appropriately label the vertices of the new figure with primes…given two figures, prove they are congruent by describing sequence of

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Coordinate plane, ordered pairs and plotting points Translation to the right or left effects the x coordinates by the same

distance and y remains the same Translation up or down effects the y coordinates by the same distance

and the x remains the same Reflection over the y axis gives the opposite x coordinate Reflection over the x axis gives the opposite y coordinate A rotation around the origin of 270 degrees gives (-y, x) clockwise A rotation around the origin of 180 degrees gives (-x, -y) clockwise A rotation around the origin of 90 degrees reverses the coordinates

and x is opposite (x, y) becomes (y, -x) clockwise A rotation of 360 degrees preserves the origin coordinates A 90 degree rotation counter-clockwise is the same as 270 degree

rotation clockwise Clockwise Counter-Clockwise Dilations produce similar figures where angle measures are preserved

and lengths are related by a constant of proportionality (scale factor) Angle sum of triangle Exterior angles of a triangle Extending a line segment of a triangle creates adjacent supplementary

angles Parallel lines Transversal Vertical Angles Supplementary Angles Corresponding angles Alternate interior angles Alternate exterior angles Protractor Use

rotations, reflections, and/or translations that would produce one from the other…on a coordinate plane, analyze and describe the coordinates of the vertices under a translation, reflection, and rotation …apply a scale factor to dilate a geometric figure. Make observations and generalizations about line segments and angles…using a composition of transformations, create a similar figure…describe the sequence of steps used to create a similar figure and state why it is similar and not congruent…analyze and describe the relationships of angles created when parallel lines are intersected by a third line (transversal)…justify angle relationships with informal arguments…physically model or use technology to prove angle sum…prove using transversals that the interior angle sum of a triangle is 180 degrees – see example…given two similar triangles created by extending the base of one. Extend the remaining sides of the triangles and show that the parallel lines created prove the two base angles are congruent…

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8.SP Statistics & Probability

Investigate patterns of association in bivariate data.

1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?KNOWStudents will know . . .


Scatter plots Bivariate data Outliers Correlations: positive, negative, or no correlation, linear or

nonlinear Line of best fit Equation of a line, slope and y intercept Categorical variables – descriptors or ranges Frequency table To calculate given quantities in percents to make comparisons To graph categorical data to create a visual for comparison

…given two sets of data (bivariate), graph one as x and one as y…make observations and fully describe the set of points…identify outliers…describe clusters in relation to the data…draw conclusions about the correlation between the sets of data…indicate the relationship as positive – both values increasing in relation to each other or negative – as one value increases the other decreases…determine if a line could be used to help describe the data…determine the best location to sketch a line that would represent the linear relationship…calculate the slope, y intercept and equation of the line of best fit…make predictions about the relationship based upon the equation of the

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line…describe the relationship between the sets of data as it relates to the line in terms of the information given (see example under standard 3)…determine the characteristics of a line that indicate a strong relationship and one that would be a weak relationship…if a correlation is non-linear, describe the line sketched through the center of the data…examine categorical data given on a frequency table…identify the independent and dependent variables…complete the chart with totals if needed and rewrite the table in terms of percent for comparison (relative frequencies)…create a bar graph as a visual of the data…analyze the table, frequencies, graph and show with evidence any correlation between the two sets of data

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Mathematics: Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levelsshould seek to develop in their students. These practices rest on important “processes and proficiencies” withlongstanding importance in mathematics education. The first of these are the NCTM process standards of problemsolving, reasoning and proof, communication, representation, and connections. The second are the strands ofmathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning,strategic competence, conceptual understanding (comprehension of mathematical concepts, operations andrelations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately),and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled witha belief in diligence and one’s own efficacy).

1. Make sense of problems and persevere in solving them.Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entrypoints to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about theform and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.They consider analogous problems, and try special cases and simpler forms of the original problem in order to gaininsight into its solution. They monitor and evaluate their progress and change course if necessary. Older studentsmight, depending on the context of the problem, transform algebraic expressions or change the viewing window ontheir graphing calculator to get the information they need. Mathematically proficient students can explaincorrespondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important featuresand relationships, graph data, and search for regularity or trends. Younger students might rely on using concreteobjects or pictures to help conceptualize and solve a problem. Mathematically proficient students check theiranswers to problems using a different method, and they continually ask themselves, “Does this make sense?” Theycan understand the approaches of others to solving complex problems and identify correspondences betweendifferent approaches.

2. Reason abstractly and quantitatively.Mathematically proficient students make sense of quantities and their relationships in problem situations. They bringtwo complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize,to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering theunits involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexiblyusing different properties of operations and objects.

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3. Construct viable arguments and critique the reasoning of others.Mathematically proficient students understand and use stated assumptions, definitions, and previously establishedresults in constructing arguments. They make conjectures and build a logical progression of statements to explorethe truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize anduse counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments ofothers. They reason inductively about data, making plausible arguments that take into account the context fromwhich the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausiblearguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings,diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or madeformal until later grades. Later, students learn to determine domains to which an argument applies. Students at allgrades can listen or read the arguments of others, decide whether they make sense, and ask useful questions toclarify or improve the arguments.

4. Model with mathematics.Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life,society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe asituation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problemin the community. By high school, a student might use geometry to solve a design problem or use a function todescribe how one quantity of interest depends on another. Mathematically proficient students who can apply whatthey know are comfortable making assumptions and approximations to simplify a complicated situation, realizingthat these may need revision later. They are able to identify important quantities in a practical situation and maptheir relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyzethose relationships mathematically to draw conclusions. They routinely interpret their mathematical results in thecontext of the situation and reflect on whether the results make sense, possibly improving the model if it has notserved its purpose.

5. Use appropriate tools strategically.Mathematically proficient students consider the available tools when solving a mathematical problem. These toolsmight include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebrasystem, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with toolsappropriate for their grade or course to make sound decisions about when each of these tools might be helpful,recognizing both the insight to be gained and their limitations. For example, mathematically proficient high schoolstudents analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errorsby strategically using estimation and other mathematical knowledge. When making mathematical models, theyknow that technology can enable them to visualize the results of varying assumptions, explore consequences, andcompare predictions with data. Mathematically proficient students at various grade levels are able to identify

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relevant external mathematical resources, such as digital content located on a website, and use them to pose or solveproblems. They are able to use technological tools to explore and deepen their understanding of concepts.

6. Attend to precision.Mathematically proficient students try to communicate precisely to others. They try to use clear definitions indiscussion with others and in their own reasoning. They state the meaning of the symbols they choose, includingusing the equal sign consistently and appropriately. They are careful about specifying units of measure, and labelingaxes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, expressnumerical answers with a degree of precision appropriate for the problem context. In the elementary grades, studentsgive carefully formulated explanations to each other. By the time they reach high school they have learned toexamine claims and make explicit use of definitions.

7. Look for and make use of structure.Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, mightnotice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapesaccording to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 ×3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can seethe 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and canuse the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shiftperspective. They can see complicated things, such as some algebraic expressions, as single objects or as beingcomposed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a squareand use that to realize that its value cannot be more than 5 for any real numbers x and y.

8. Look for and express regularity in repeated reasoning.Mathematically proficient students notice if calculations are repeated, and look both for general methods and forshortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the samecalculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculationof slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school studentsmight abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of ageometric series. As they work to solve a problem, mathematically proficient students maintain oversight of theprocess, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the disciplineof mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and

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expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, andprofessional development should all attend to the need to connect the mathematical practices to mathematicalcontent in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectationsthat begin with the word “understand” are often especially good opportunities to connect the practices to the content.Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from whichto work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions,apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain themathematics accurately to other students, step back for an overview, or deviate from a known procedure to find ashortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematicalpractices.

In this respect, those content standards which set an expectation of understanding are potential “points ofintersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. Thesepoints of intersection are intended to be weighted toward central and generative concepts in the school mathematicscurriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve thecurriculum, instruction, assessment, professional development, and student achievement in mathematics.

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